= Solution
Use $k_{\rm B}T_p=1000$ eV and $k_{\rm B}T_b=100$ eV. The nonrelativistic beam speed and cross-sectional area are
$$
V=\sqrt{\frac{2(10\,\mathrm{keV})}{m_e}}\simeq5.93\times10^7\,\mathrm{m\,s^{-1}},\qquad A_b=\pi(0.5\,\mathrm{mm})^2.
$$
Here $V/\sigma_p=\sqrt{20}$, $\sigma_b^2/\sigma_p^2=0.1$, and the exponential in the requested threshold is $e^{-10}$. With $C=\sqrt e$,
$$
\left(\frac{n_b}{n_p}\right)_{\rm est}=\sqrt e\,(\sqrt{20})(0.1)e^{-10}\simeq3.35\times10^{-5}.
$$
Converting beam density to the magnitude of its <electric current> gives
$$
\boxed{I_{\min,\rm est}=en_bVA_b\simeq2.5\times10^4\,\mathrm A.}
$$
This is an order-of-magnitude estimate from the source's asymptotic formula. The numerical hierarchy is only moderate: $V\sigma_b/\sigma_p^2=\sqrt2$. Retaining the background slope at $V-\sigma_b$ instead gives about $9.1\times10^4$ A as a sufficient positive-slope current at that chosen velocity. The difference quantifies the sensitivity of the exponential-tail approximation; an exact kinetic instability threshold requires the dispersion relation. The requested estimate is therefore on the scale of tens of kiloamperes, rather than a precisely determined current.
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